Optimal sensing of photon addition and subtraction on nonclassical light

TL;DR

The paper addresses detecting photon addition and subtraction in nonclassical light directly from optical tomograms using the Wasserstein distance $W_{1}$, avoiding full state tomography. It formalizes the tomogram $w(X_{ heta}, \theta)$ and computes $W_{1}$ between tomograms by integrating differences of their CDFs, emphasizing that $W_{1}$ is a true metric sensitive to interference patterns. Results for the squeezed vacuum state $|\xi\rangle$ and its photon-added/subtracted variants reveal that photon-number changes induce quadrature-dependent shifts and fringe rearrangements in the tomogram, with crossovers in $W_{1}$ as a function of the squeezing parameter $r$. The study also highlights tomographic connections between SVS and ECS, demonstrates reconstruction-free state discrimination, and discusses extensions to multimode states and metrological applications.

Abstract

We demonstrate that the Wasserstein distance $W_{1}$ corresponding to optical tomograms of nonclassical states faithfully captures changes that arise due to photon addition to, or subtraction from, these states. $W_{1}$ is a true measure of distance in the quantum state space, and is sensitive to the underlying interference structures that arise in the tomogram after changes in the photon number. Our procedure is universally applicable to the cat and squeezed states, the former displaying the characteristic negativity in its Wigner function, while the latter does not do so. We explicate this in the case of the squeezed vacuum and even coherent states and show that photon addition (or subtraction) is mirrored in the shift in the intensity of specific regions in the tomogram. Further, we examine the dependence of $W_{1}$ on the squeezing parameter, and its sensitivity to different quadratures.

Figures (3)

• Figure 1: Left to right: Single-mode optical tomograms corresponding to (a) $|\xi\rangle$, (b) $|\xi,1\rangle$ (or $|\xi,-1\rangle$), (c) $|\xi,2\rangle$, (d) $|\xi,3\rangle$, (e) $|\xi,-2\rangle$ and (f) $|\xi,-3\rangle$. The squeezing parameter $\xi = re^{i\phi}$ ($r = 1/\sqrt{2}$ and $\phi = 0$). The bright central region at $\theta=0$, $\pi$, $2\pi$ in (a) indicates a peak in the PDF. Addition or subtraction of a single photon to the SVS $|\xi\rangle$ destroys this peak (the central dark region in (b)). With the addition or subtraction of two photons the bright central region reappears ((c) and (e)). With the addition or subtraction of three photons the bright central region disappears, and is replaced by a central dark region ((d) and (f)).
• Figure 2: Probability distribution function (PDF) along the $x$-quadrature corresponding to $|\xi\rangle$ (blue), $|\xi,1\rangle$ (orange), and $|\xi,2\rangle$ (green) for $r =$ (a) $0.38$ and (b) $0.52$. The other parameter values are as in Fig. \ref{['fig:fig_tomogram_SVS_PASVS_PSSVS_DPASVS_TPASVS_DPSSVS_TPSSVS_r_1OverSqrt2_phi_0_panel']}. At $x = 0$, the PDF for the SVS $|\xi\rangle$ and the two-photon added SVS $|\xi,2\rangle$ are maximum, while the one-photon added SVS $|\xi,1\rangle$ has a corresponding minimum. This is reflected in the shift in the intensity patterns seen in Fig. \ref{['fig:fig_tomogram_SVS_PASVS_PSSVS_DPASVS_TPASVS_DPSSVS_TPSSVS_r_1OverSqrt2_phi_0_panel']}.
• Figure 3: Wasserstein distance $W_{1}$ between $|\xi\rangle$ and $|\xi, 1\rangle$ [or $|\xi,-1\rangle$] (black circles), $|\xi\rangle$ and $|\xi, 2\rangle$ (red triangles), and $|\xi\rangle$ and $|\xi, 3\rangle$ (green asterisks) as a function of the squeezing parameter $r$, along the $x$-quadrature. The other parameters are as in Fig. \ref{['fig:fig_tomogram_SVS_PASVS_PSSVS_DPASVS_TPASVS_DPSSVS_TPSSVS_r_1OverSqrt2_phi_0_panel']}. The crossover between $W_1$ for one photon addition and two photon addition is at $r \approx 0.45$ and that between one photon addition (the same as one photon subtraction) and three photon addition is at $r \approx 0.59$. To identify the extent of photon addition and to distinguish between photon added counterparts using $W_{1}$ it is therefore important to avoid the neighborhood of values of $r$ where crossovers occur.